# 12 Days of Christmas puzzles from Oxford professor to keep you on your toes over the hols

If you’re looking for some to test your guests over Christmas dinner this year, then why not try on of these festive puzzles.

Dr Tom Crawford, who recently explained how many legs are in the 12 days of Christmas, has put together 12 head-catchers to get you thinking in the run up to the big day.

We caught up with Tom to find out why he feels it’s important to keep people’s minds sharp even during the jolly season.

Tom said: “As a huge fan of both maths and Christmas, creating the 12 Days of Christmas puzzles was a real delight.

“The idea behind the puzzles is to try to encourage people to engage with maths and to show them that it can be a lot more fun than they realised when at school.

“Being able to work with numbers is an essential tool in everyday life, from staying on top of your finances, to understanding the statistics being quoted in the news, maths helps you to make better decisions.

“Hopefully, by having a go at these Christmas puzzles you’ll not only sharpen your mind, but also get a great sense of achievement when you work out the right answer.”

So without further ado, here are some questions to wrap your head around over the 12 days of Christmas, courtesy of Dr Tom Crawford (the answers can be found at the bottom of this article).

## 12 days of Christmas puzzles

Puzzle one: If I set a puzzle every day of the advent period (1-25 December) and spend one minute on the first puzzle, two minutes on the second, three minutes on the third, and so on, with the final one being 25 minutes on the 25th puzzle, what is the total amount of time I will spend writing puzzles?

Puzzle two : 6 December marked my birthday and to celebrate I travelled to Kiev with four friends. If I order a drink on the flight out and then each of my friends orders twice as many as the person before, how many drinks do we order in total?

Puzzle three: This morning I built a snowman using three spheres of radius 0.5m, 0.4m and 0.2m. However, the sun has since come out and the snowman is starting to melt at a rate of 0.01 m3 per minute. How long will it take for him to disappear completely?

Puzzle four: Suppose a newly-born pair of elves, one male, one female, are living together at the North pole. Elves are able to mate at the age of one month so that at the end of its second month a female elf can produce another pair of offspring. Suppose that the elves never die, and that the female always produces one new pair (one male, one female) every month from the start of the third month on. After one year, how many pairs of elves will there be?

Puzzle five: On Christmas day, I have 11 people coming to dinner and so I’m working on the seating plan ahead of time. For a round table with exactly 12 chairs, how many different seating plans are possible?

Puzzle six: My front yard is covered in snow and I need to clear a path connecting my front door to the pavement and then back to the garage. If each square in the diagram is 1m x 1m what is the shortest possible path?

Puzzle seven: The first night of Chanukah is 22 December when the first candle is lit. If it burns at a rate of 0.05cm per hour, how tall does the candle need to be to last the required eight days?

Puzzle eight: If you have a square chimney which is 0.7m across, assuming Santa has a round belly, what is the maximum waist size that can fit down the chimney?

Puzzle nine: On Christmas Eve Santa needs to visit each country around the world in 24 hours. Assuming time stands still whilst he is travelling, how long can he spend in each country?

Puzzle 10: I got carried away with buying presents this year and now have more than can fit into my stocking. If the stocking has a maximum capacity of 150, and my presents have the following sizes: 16, 27, 37, 65, 52, 42, 95, 59; what is the closest I can get to filling the stocking completely?

(NB: The question is not looking for the highest number of presents that will fit, but rather the largest total that is less than or equal to 150).

Puzzle 11: Santa has eight reindeer, and each one can pull a weight of 80kg. If Santa weights 90kg, his sleigh 180kg, and each present weighs at least 3kg, what is the maximum number of presents that can be carried in a single trip?

Puzzle 12: To mark the end of the 12 days of Christmas each student at the University of Oxford has kindly decided to donate some money to a charity of their choice. If the first person donates £12 and everyone after donates exactly half the amount of the person before them (rounding down to the nearest penny), how much will be donated in total?

Caution, spoilers ahead! Once you’ve given your brain a run for its money (no peeking), here are the answers you’ve been waiting for, as explained by Tom himself.

Puzzle one: 1 + 2 + 3 + … + 25 = 325. There is a faster way to do this which was first discovered by the mathematician Gauss when he was still at school. If you pair each of the numbers in your sum, eg. 0 + 25, 1 + 24, 2 + 23, etc. up to 12 + 13, then you have 13 pairs which each total 25 and so the overall total is 25*13 = 325. The same method works when adding up the first n numbers, with the total always being n(n+1)/2.

Puzzle two: 1+2+4+8+16 = 31.

Puzzle three: Volume of a sphere = (4/3)*pi*radius3 and so the total volume of snow = 0.52 + 0.27 + 0.03 = 0.82 m3. Melting at a rate of 0.01 m3 per minute means the snowman will be gone after only 82 minutes!.

Puzzle four: This problem is actually an incredibly famous sequence in disguise…

The first new pair is born at the start of the third month giving two pairs after three months. The question tells us that we have to wait one whole month before the new offspring can mate and so only the original pair can give birth during the fourth month which leaves a total of three pairs after four months. For the fifth month, both the original pair, and the first-born pair can now produce offspring and so we get two more pairs giving a total of five after five months. In month six, the second-born pair can now also produce offspring and so in total we have three offspring-producing pairs, giving eight pairs after six months.

At this point, you may have spotted that the numbers follow the Fibonacci sequence, which is created by adding the previous two numbers together to get the next one along. The first twelve numbers in the sequence are below, which gives an answer of 144.

Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Puzzle five: I have 12 choices of where to place the first person, 11 for the second, 10 for the third and so on, which gives 12*11*10*9*8*7*6*5*4*3*2*1 = 12 (read as 12 factorial) in total. But, for any given seating plan, we can rotate around the table one place to get the same order, which means we have in fact over counted by a factor of 12. Therefore, the total number is 11 = 39,916,800.

Puzzle six: Reflect the yard in the pavement and draw a straight line connecting the front door to the edge of the garage closest to the front door (blue). Then add the same line from the ‘reflected’ front door at the top back down to the garage at the bottom (orange). The final shortest path is found by combining both paths for a valid one in the original diagram.

The length is found using Pythagoras’ Theorem. From the door to the pavement we have length

(12 + 22)1/2 = (5)1/2

and from the pavement to the garage the length is

((1.5)2 + 32)1/2 = (11.25)1/2

giving a total length of 2.23 + 3.35 = 5.58m.

Puzzle seven: Eight days = 8*24 hours = 192 hours. 192*0.05 = 9.6cm.

Puzzle eight: Chimney diameter = 0.7m so the maximum circumference (or waist size) that will fit is 0.7*pi = 2.2m or 88 inches.

Puzzle nine: Using the UN list of 193 countries, Santa has 24 * 60 = 1440 minutes total, which means spending only 7.5 minutes in each country.

Puzzle 10: 150 exactly with 16 + 27 + 42 + 65 = 150.

Puzzle 11: We have eight reindeer each with a capacity of 80kg giving a total of 640kg that can be carried. Subtracting the 90kg for Santa and 180kg for the sleigh leaves 370kg available. Dividing this by 3 gives 123.33 so a maximum of 123 presents can be carried at once.

Puzzle 12: 12 + 6 + 3 + 1.5 + 0.75 + 0.37 + 0.18 + 0.09 + 0.04 + 0.02 + 0.01 + 0 + 0 + 0 + …

The donations stop after the 11th person giving a total of £23.87. Even if we had allowed donations of part of a penny the total would never quite reach £24.00. This is an example of an infinite sum (or Geometric Series) where the total is always two times the first number.